Operations on Matrices

•    Let A = [aij] m x n and B = [bij] m x n be two matrices of same order m x n. Then the sum of the two matrices A and B is defined as a matrix
C = [cij] m x n  where cij = aij + bij   for all possible values of i and j.

•    If A and B are matrices of same order, then A + B =  B + A. This is commutative property of matrices under addition. Addition of matrices also holds the associative law.

•    For any matrix A, there exists a null matrix O of the same order such that A + O = A = O + A.

•    For any matrix A, there exists a matrix  – A such that:
•    A + (– A) = O = (–A) + A
•    Two matrices are subtracted by subtracting corresponding  elements of these matrices.
•    Let A = [aij] m x n is a matrix and k is a scalar, then kA is another matrix, which is obtained by multiplying each element of A by a scalar k.

•    The product AB of two matrices A and B is defined if numbers of columns of A is equal to the number of rows of B.   

•    Matrix multiplication is not commutative in general.

•    Matrix multiplication is associative whenever both sides of equality are defined.

•    Matrix multiplication is distributive over matrix addition.

•    For every square matrix A, there is an identity matrix of same order such that IA = AI = A.

•    The product of two matrices can be null matrix while neither of them is a null matrix. If A is an m × n matrix and O is a null matrix, then
•    the product of matrix A with the null matrix O is always a null matrix.

•    In case of matrix multiplication if AB = O, then it does not necessarily imply that BA = O.

•    The transpose of a matrix is obtained by interchanging its rows and columns. It is denoted by  A' or  AT.
•    Properties of Transpose:

•    For any square matrix A with real number entries, A + AT is a symmetric matrix and A – AT is a skew symmetric matrix.

•    Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.


Keywords: Addition of matrices, Multiplication of matrices, Difference of matrices, Transpose of a matrix, Multiplication of a matrix by a scalar, Symmetric matrix, Skew symmetric matrix

 

 

 


 

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  • Q1

    If A and B are two matrices such that A + B and AB are both defined, then

    Marks:1
    Answer:

    A and B are square matrices of same order.

    Explanation:
    In square matrices, both addition and multiplication are defined. Therefore, A and B are square matrices of same order.
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  • Q2

    If A is a square matrix, then AAT is always a

    Marks:1
    Answer:

    symmetric matrix.

    Explanation:

    (AAT )T = (AT )T A= AAT  

    Hence AAT is a symmetric matrix.

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  • Q3

    Marks:1
    Answer:

    Explanation:

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  • Q4

    If A and B are symmetric matrices of the same order, then (AB – BA) is

    Marks:2
    Answer:

    A skew-symmetric matrix

    Explanation:

    Let A and B be symmetric.

    Then At = A and Bt = B

    (AB – BA)t = (AB)t – (BA)t

    = BtAt – AtBt = BA – AB = – (AB – AB)

    Hence, (AB – BA) is skew-symmetric.

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  • Q5

    If A is skew-symmetric, then A3 is :

    Marks:1
    Answer:

    skew-symmetric

    Explanation:

    Let A be skew symmetric

    Then, At = – A

    (A3)t = (AAA)t = At. At. At

            = (–A) (–A) (–A) = – A3

    Hence, A3 is skew-symmetric

    View Answer